Completed cohomology and the -adic Lang- lands...

Completed cohomology and the p-adic Lang-

lands program

Matthew Emerton ∗

Abstract. We discuss some known and conjectural properties of the completed coho-mology of congruence quotients associated to reductive groups over Q. We also discussthe conjectural relationships to local and global Galois representations, and a possiblep-adic local Langlands correspondence.

Mathematics Subject Classification (2010). Primary 11F70; Secondary 22D12.

Keywords. p-adic Langlands program, p-adic Hodge theory, completed cohomology,

Galois representations.

1. Introduction

The idea that there could be a p-adic version of the local Langlands correspondencewas originally proposed by C. Breuil, in the case of the group GL2(Qp), in hispapers [11, 12, 13]. In [13], he also proposed a local-global compatibility betweenthis (at the time conjectural) correspondence and p-adically completed cohomologyof modular curves. Since then, the theory of p-adic Langlands has been extensivelydeveloped in the case of the group GL2 over Q; see e.g. [6, 29, 50, 30, 32, 46, 52].Since excellent expositions of much of this material already exist [5, 14, 15], wehave decided here to describe some aspects of the p-adic Langlands program thatmake sense for arbitrary groups. This has led us to focus on completed cohomology,since this is a construction that makes sense for arbitrary groups, and about whichit is possible to establish some general results, and make some general conjectures.

Many of these conjectures are very much motivated by the conjectural relation-ship with Galois representations, and we have also tried to outline our expectationsregarding this relationship, while trying not get bogged down in the myriad tech-nical details that would necessarily attend a more careful discussion of this topic.Finally, we have tried to indicate how completed cohomology may be related, bythe principle of local-global compatibility, to a still largely conjectural p-adic lo-cal Langlands correspondence for groups other than GL2(Qp). Our focus is ondrawing inferences about the possible structure of the local correspondence by in-terpolating from our expectations regarding completed cohomology. In this regardwe mention also the paper [27], which somewhat literally interpolates p-adically

∗The author’s work is partially supported by NSF grant DMS-1303450

2 Matthew Emerton

completed cohomology (via the Taylor–Wiles–Kisin method) in order to constructa candidate for the p-adic local Langlands correspondence for GLn of an arbitraryp-adic field.

We close this introduction by simply listing a number of additional importantpapers [17, 18, 19, 20] which also have the goal of extending the p-adic local Lang-lands correspondence, and local-global compatibility, beyond the case of GL2(Qp).Unfortunately, lack of space prevents us from saying more about them here.

1.1. Notation. Throughout, we fix a prime p. We also fix a finite extensionL of Qp, contained inside some given algebraic closure Qp. We let O denote thering of integers of L, and let $ denote a uniformizer of O. The ring O, and field L,will serve as our coefficients.

As usual, A denotes the ring of adeles over Q, Af denotes the ring of finiteadeles, and Ap

f denotes the ring of prime-to-p finite adeles. For a given algebraicgroup G over Q, we will consistently write G∞ := G(R), and G := G(Qp). Addi-tional notation will be introduced as necessary.

1.2. Acknowledgements. The ideas expressed here owe much to the manydiscussions with my colleagues and collaborators that I’ve enjoyed over the years,and I would like to thank them, especially Ana Caraiani, Pierre Colmez, DavidGeraghty, Michael Harris, David Helm, Florian Herzig, Mark Kisin, David Savitt,and Sug Woo Shin. I am particularly indebted to Christophe Breuil, whose ideasregarding the p-adic Langlands program have been so influential and inspiring.Special thanks are owed to Frank Calegari, Toby Gee, and Vytas Paskunas; manyof the ideas described here were worked out in collaboration with them.

I am also grateful to Frank Calegari, Tianqi Fan, Toby Gee, Michael Harris,Daniel Le, David Savitt, and the Imperial College Study Group (Rebecca Bellovin,Kevin Buzzard, Toby Gee, David Helm, Judith Ludwig, James Newton, and JackShotton) for their careful reading of various preliminary versions of this note.

2. Completed cohomology

Ideally, in the p-adic Langlands program, we would like to define spaces of p-adicautomorphic forms, to serve as p-adic analogues of spaces of classical automorphicforms in the usual Langlands program. Unfortunately, for general reductive groups,no such definition is currently available. For groups that are compact at infinity,we can make such a definition (see (2.2.4) below), while for groups giving riseto Shimura varieties, we can use algebro-geometric methods to define spaces ofp-adic automorphic forms, which however seem less representation-theoretic innature than classical automorphic forms. Since one of our main goals is to employrepresentation-theoretic methods, we are thus led to find an alternative approach.

The approach we take here is to work with p-adically completed cohomology.This has the advantages of being definable for arbitrary groups, and of beingof a representation-theoretic nature. For our purposes, it will thus serve as a

Completed cohomology and the p-adic Langlands program 3

suitable surrogate for a space of p-adic automorphic forms. For groups that arecompact at infinity, it does recover the usual notion of p-adic automorphic formsthat was already mentioned. (Its relationship to algebro-geometric notions of p-adic automorphic forms in the Shimura variety context is less clear; P. Scholze’spaper [53] makes fundamental progress in this — and many other! — directions,but we won’t attempt to discuss this here.) We therefore begin our discussionof global p-adic Langlands by recalling the basic definitions and facts related tocompleted cohomology, referring to [24] and [31] for more details.

2.1. Definitions and basic properties. Let us suppose that G is areductive linear algebraic group over Q. We write G∞ := G(R) for the group ofreal-valued points of R; this is a reductive Lie group. We let A∞ denote the R-points of the maximal Q-split torus in the centre of G, and let K∞ denote a choiceof maximal compact subgroup of G∞. For any Lie group H, we let H◦ denote thesubgroup consisting of the connected component of the identity.

The quotient G∞/A◦∞K◦∞ is a symmetric space on which G∞ acts. We denote

its dimension by d. We also write l0 := rank of G∞ − rank of A∞K∞, andq0 := (d − l0)/2. If G∞ is semisimple (so that in particular A∞ is trivial) thenthese quantities coincide with the quantities denoted by the same symbols in [10](which is why we notate them as we do). We note that q0 is in fact an integer.(For the role played by these two quantities in our discussion, see (2.1.6), as wellas Conjectures 3.1 and 3.2, below.)

If Kf ⊂ G(Af ) is a compact open subgroup, then we write

Y (Kf ) := G(Q)\G(A)/A◦∞K◦∞Kf .

The double quotient Y (Kf ) is a finite union of quotients of the symmetric spaceG◦∞/A◦∞K◦

∞ by cofinite volume discrete subgroups of G∞.

2.1.1. Definitions. We fix a tame level, i.e. a compact open subgroup Kpf ⊂

G(Apf ). If Kp ⊂ G := G(Qp) is a compact open subgroup, then of course KpK

pf is

a compact open subgroup of G(Af ), and so we may form the space Y (KpKpf ). We

define the completed (co)homology at the tame level Kpf as follows [24]:

Hi := lim←−s

lim−→Kp

Hi(Y (KpK

pf ),O/$s

)and Hi := lim←−

Kp

Hi

(Y (KpK

pf ),O

), (1)

where in both limits Kp ranges over all compact open subgroups of G(Qp).We equip each of Hi and Hi with its evident inverse limit topology. In the

case of Hi, each of the O/$s-modules lim−→KpHi

(Y (KpK

pf ),O/$s

)appearing in

the projective limit is equipped with its discrete topology, and the projective limittopology on Hi coincides with its $-adic topology; Hi is complete with respectto this topology. In the case of Hi, each of the terms Hi

(Y (KpK

pf ),O

)(which is

a finitely generated O-module) appearing in the projective limit is equipped withits $-adic topology. The topology on Hi is then a pro-finite topology, and so inparticular Hi is compact.

4 Matthew Emerton

The completed cohomology and homology are related to one another in theusual way by duality over O [24]. In particular, if we ignore O-torsion, then Hi

is the O-dual of Hi (equipped with its weak topology), and Hi is the continuousO-dual of Hi.

We can also form the limit

Hi := lim−→Kp

lim←−s

Hi(Y (KpK

pf ),O/$s

) ∼= lim−→Kp

Hi(Y (KpK

pf ),O

). (2)

There is a natural morphismHi → Hi, (3)

which induces an embeddingHi ↪→ Hi, (4)

whose source is the $-adic completion of Hi. Note that the morphism (3) neednot be injective: although each of the terms Hi

(Y (KpK

pf ),O

)in the direct limit

defining Hi is a finitely generated O-module, the limit Hi is merely countablygenerated as an O-module, and hence may contain divisible elements. These thenbecome zero after we pass to the $-adic completion to obtain the embedding (4).(We give an example of this in (2.2.3) below.)

We furthermore remark that the transition maps in the direct limits (1) and (2)need not be injective. (This is also illustrated by the example of (2.2.3).) However,if K ′

p ⊂ Kp, then a trace argument shows that the restriction map

Hi(Y (KpK

pf ),O)→ Hi

(Y (K ′

pKpf ),O)

becomes injective after tensoring with L. Thus the kernel of the natural mapHi

(Y (KpK

pf ),O)→ Hi consists of torsion classes.

On the other hand, the kernel of the natural map

Hi(Y (KpK

pf ),O)→ Hi (5)

need not consist only of torsion classes; it is possible for non-torsion classes to haveinfinitely divisible image in Hi, and hence have vanishing image in Hi (and so alsohave vanishing image in Hi). (Again, we refer to (2.2.3) for an example of this.).

The morphism (4), although injective, need not be surjective. Its cokernel isnaturally identified with TpH

i+1 := lim←−sHi+1[$s]; in other words we have a short

exact sequence0→ Hi → Hi → TpH

i+1 → 0. (6)

(Here we have written Hi+1[$s] to denote the submodule of $s-torsion elementsin Hi+1, and the projective limit is taken with respect to the multiplication-by-$map from Hi+1[$s+1] to Hi+1[$s]. The notation “Tp” is for Tate module.) Inparticular, if all the cohomology modules Hi+1

(Y (KpK

pf ),O

)are torsion free, so

that Hi+1 is torsion free, then the morphism (4) is an isomorphism.Although the restriction maps (5) can have non-trivial kernels, one can recover

the cohomology at the various finite levels KpKpf via the Hochschild–Serre spectral


sequence discussed in (2.1.3) below. The precise manner in which the cohomologyat finite levels gets encoded in the completed cohomology is somewhat complicatedin general. For instance, infinitely divisible torsion elements in Hi give rise toelements of TpH

i, which by the discussion of the previous paragraph is naturallya quotient of Hi−1. However, infinitely divisible non-torsion elements of Hi haveno obvious incarnation as elements of completed cohomology, and the manner inwhich they are recovered by the Hochschild–Serre spectral sequence can be subtle.

2.1.2. Group actions. There is a natural continuous action of G := G(Qp) oneach of Hi and Hi, whose key property is encapsulated in the following result.

Theorem 2.1. [24, 31] The G-action on Hi makes it a $-adically admissiblerepresentation of G; i.e. it is $-adically complete as an O-module, and each quo-tient Hi/$s (which is a smooth representation of G over O/$s) is admissiblein the usual sense (for each compact open subgroup Kp ⊂ G, the submodule ofKp-invariants is finitely generated over O).

If Kp ⊂ G is compact open, then we write O[[Kp]] := lim←−K′p

O[Kp/K ′p], where

the projective limit is taken over all open normal subgroups K ′p ⊂ Kp. Since G

acts continuously on Hi and Hi, it follows that for any such Kp, the Kp-action oneach of these modules may be promoted to an action of O[[Kp]]. One then has thefollowing reformulation of the admissibility of the G-action on Hi.

Theorem 2.2. Each of the O[[Kp]]-modules Hi is finitely generated.

(We remark that if this finite generation statement holds for one choice of Kp

then it holds for any such choice.)

2.1.3. Cohomology of local systems and the Hochschild–Serre spectralsequence. If W is a finitely generated O-module equipped with a continuousrepresentation of an open subgroup Kp ⊂ G, and Kp is sufficiently small, sothat G(Q) acts with trivial stabilizers on G(A)/A◦∞K◦

∞KpKpf , then, for each open

subgroup K ′p ⊂ Kp, the representation W determines a local system W of O-

modules on Y (KpKpf ), defined via W := G(Q)\

((G(Af )/A◦∞K◦

∞Kpf

)×W

)/K ′

p.

Suppose that W is furthermore torsion-free as an O-module, and let W∨ denote theO-dual of W , endowed with the contragredient Kp-action. There is then [24, 31]a Hochschild–Serre spectral sequence

Ei,j2 = Exti

O[[Kp]](W∨, Hj) =⇒ Hi+j

(Y (KpK

pf ),W

).

This gives a precise sense to the idea that p-adically completed cohomology cap-tures all the cohomology (with arbitrary coefficients) at finite levels. It is therealization, in the context of completed cohomology, of the general philosophy(brought out especially in Hida’s work, e.g. [44]) that when working with p-adicautomorphic forms, by passing to infinite p-power level one automatically encom-passes automorphic forms of all possible weights. (For the precise relationship withclassical automorphic forms, see (2.1.6) below.)

6 Matthew Emerton

In particular, if we take W = O (with the trivial Kp-action), we obtain aspectral sequence Ei,j

2 = Hi(Kp, H

j)

=⇒ Hi+j(Y (KpK

pf ),O

)(where Hi denotes

continuous cohomology), which recovers the cohomology at the finite level KpKpf

from the completed cohomology. If we take a direct limit over all Kp, we obtain aspectral sequence Ei,j

2 = lim−→KpHi

(Kp, H

j)

=⇒ Hi+j , which recovers the limits

Hi+j of the cohomology at finite levels. The edge map Hi → lim−→Kp(Hi)Kp is

induced by the morphism (3). As already noted, the relationship between thisspectral sequence, the morphism (3), and the exact sequence (6) is subtle. (Seethe example of (2.2.3) below.)

More generally, if we write Hi(W) := lim−→K′p

Hi(Y (K ′

pKpf ),W

)(where K ′

p runs

over all open subgroups of the fixed Kp, of which W is a representation), then wehave an edge map Hi(W)→ lim−→K′

p

HomK′p(W∨, Hi

)which relates the cohomology

at finite levels with coefficients inW to the W∨-isotypic parts (for open subgroupsK ′

p of Kp) of Hi.

2.1.4. Hecke actions. There is a finite set of primes Σ0 (containing p) such thatfor ` 6∈ Σ0, we may factor Kp

f = Kp,`f K` where Kp,`

f is compact open in G(Ap,`f ),

and K` is a hyperspecial maximal compact subgroup of G(Q`). (In particular, G isunramified at such a prime `, so that it admits a hyperspecial maximal compactsubgroup.) We may then consider the spherical Hecke algebra (i.e. the doublecoset algebra) H` := H

(G(Q`)//K`,O

)with coefficients in O. This algebra is

commutative [41, 42], and acts naturally (by continuous operators) on any of thecohomology groups Hi

(Y (KpK

pf ),W) considered in (2.1.3).

If we let i range over all cohomological degrees, let Kp range over all compactopen subgroups of G, and let W range over all representations of Kp on finitelygenerated torsion O-modules, then

∏i

∏Kp

∏W EndHi

(Y (KpK

pf ),W) is a profi-

nite ring. We fix a set of primes Σ containing Σ0, and we define the global Heckealgebra TΣ to be the closure in this profinite ring of the O-subalgebra generated bythe image of H` for all ` 6∈ Σ. (The reasons for allowing the possibility of Σ beinglarger than Σ0 are essentially technical; it is not misleading to simply imagine thatΣ is equal to Σ0 and is fixed once and for all, e.g. by being taken to be as smallas possible, given our fixed choice of tame level Kp

f .) By construction, TΣ actson each of the cohomology groups Hi

(Y (KpK

pf ),W) considered in (2.1.3) (this is

obvious if W is torsion, and follows in general by writing W as a projective limitW ∼= lim←−s

W/$s), and also on Hi and Hi. (In the case of Hi this follows from itsdefinition in terms of limits of cohomology groups Hi

(Y (KpK

pf ),O/$s

). It then

follows for Hi by duality; more precisely, we can write Hi as the projective limitover s and Kp of Hi

(Y (KpK

pf ),O/$s

), and this latter module is the O/$s-dual

of Hi(Y (KpK

pf ),O/$s

).)

These Hecke actions on Hi and Hi commute with the action of G. Thusthere is an action of Hecke on the E2-terms, as well the E∞-terms, of the variousHochschild–Serre spectral sequences considered in (2.1.3), and these are in fact


spectral sequences of TΣ-modules.The O-algebra TΣ is a complete semi-local ring, i.e. it is a product of finitely

many factors Tm, each of which is a complete local ring. (This finiteness statementis proved via the methods of [3].) The topology on TΣ is the product of the m-adictopologies on each of the complete local rings Tm.

2.1.5. Variants in the non-compact case, and Poincare duality. Thereare some variants of completed (co)homology which are useful in the case whenthe quotients Y (Kf ) are non-compact. (We recall that the quotients Y (Kf ) arecompact precisely when the semisimple part of the group G is anisotropic, i.e.contains no torus that is split over Q.)

Namely, replacing cohomology by cohomology with compact supports, we candefine G-representations Hi

c, Hic, Hi

c, and TpHic, and we have Hochschild–Serre

spectral sequences for compactly supported cohomology. Similarly, we can definecompleted Borel–Moore homology HBM

i , which is related to Hic by duality over O.

There is a more subtle duality over O[[Kp]] (for a compact open subgroupKp ⊂ G) which relates the usual and compactly supported variants of completedcohomology. It is most easily expressed on the homology side, where it takes theform of the Poincare duality spectral sequence

Ei,j2 := Exti

O[[Kp]](Hj ,O[[Kp]]) =⇒ HBMd−(i+j). (7)

Of course, when the quotients Y (Kf ) are compact, compactly supported andusual cohomology coincide, as do usual and Borel–Moore homology. In general,we can relate them by considering the Borel–Serre compactifications Y (Kf ). If welet ∂(Kf ) denote the boundary of Y (Kf ), then we may compute the compactlysupported cohomology of Y (Kf ) as the relative cohomology

Hic

(Y (Kf ),O/$s

)= Hi

(Y (Kf ), ∂(Kf );O/$s

)(and similarly we may compute Borel–Moore homology as relative homology),and so we obtain the long exact sequence of the pair relating the cohomology ofY (Kf ), the compactly supported cohomology of Y (Kf ), and the cohomology of∂(Kf ). Passing to the various limits, we obtain a long exact sequence

· · · → Hic → Hi → Hi(∂) := lim←−

s

lim−→Kp

Hi(∂(KpK

pf ),O/$s

)→ Hi+1

c → · · · .

The completed cohomology of the boundary Hi(∂) can be computed in terms ofcompleted cohomology of the various Levi subgroups of G; see [24] for more details.

We also mention that there is a TΣ-action on each of the objects introducedhere, and that the various spectral sequences and long exact sequences consideredhere are all compatible with these actions.

2.1.6. The relationship to automorphic forms. We fix an isomorphism ı :Qp∼= C. Since L ⊂ Qp, we obtain induced embeddings O ⊂ L ↪→ C.

8 Matthew Emerton

Suppose that V is an algebraic representation of G defined over L, and supposefurther that W is a Kp-invariant O-lattice in V, for some compact open subgroupKp of G := G(Qp) ⊂ G(L). Write VC := C⊗L V = C⊗O W (the tensor productsbeing taken with respect to the embeddings induced by ı). If, for any compactopen subgroup K ′

p of Kp, we let VC denote the local system of C-vector spaces onY (K ′

pKpf ) associated to VC, then we obtain a natural isomorphism

C⊗O Hi(W) ∼= lim−→K′

p

Hi(Y (K ′

pKpf ),VC

)=: Hi(VC).

We may compute this cohomology using the space of automorphic forms on G [39].Precisely, write A(Kf ) for the space of automorphic forms on G(Q)\G(A)/Kf ,

for any compact open subgroup Kf ⊂ G(Af ), and write A(Kpf ) = lim−→Kp

A(KpKpf ),

where the direct limit is taken (as usual) over the compact open subgroups Kp ⊂G := G(Qp). For simplicity, suppose that A◦∞ acts on VC through some characterχ (this will certainly hold if V is absolutely irreducible), and let A(Kp

f )χ−1 denotethe subspace of A(Kp

f ) on which A◦∞ acts through the character χ−1. Let G∞denote the group of real points of the intersection of the kernels of all the rationalcharacters of G, and let g and k denote the Lie algebras of G∞ and K∞ respectively.Then it is proved by J. Franke in [39] that there is a natural isomorphism

Hi(VC) ∼= Hi(g, k;A(Kp

f )χ−1 ⊗ VC). (8)

(In the case when the the quotients Y (Kf ) are compact, this result is known asMatsushima’s formula [49]. In the case when G = GL2 and i = 1, it is knownas the Eichler–Shimura isomorphism [54].) The action of TΣ on Hi(W) inducesan action of TΣ on Hi(VC), and the resulting systems of Hecke eigenvalues thatappear in Hi(VC) (which we may think of as being simultaneously C-valued andQp-valued, by employing the isomorphism ı) are automorphic, i.e. arise as systemsof Hecke eigenvalues on the space of automorphic forms A(Kp

f ).Thus, to first approximation, we may regard the Hecke algebra TΣ (or, more

precisely, the Qp-valued points of its Spec) as being obtained by $-adically interpo-lating the automorphic systems of Hecke eigenvalues that occur in Hi. However, aswe already noted, the map from Hi to Hi can have a non-trivial kernel. More signif-icantly (given that the Hochschild–Serre spectral sequence shows that, despite thispossibility, the systems of Hecke eigenvalues appearing in Hi can be recovered fromthe completed cohomology), the inclusion Hi ↪→ Hi can have a non-trivial cokernelTpH

i+1, which arises from infinitely divisible torsion in Hi+1. Thus Spec TΣ seesnot only the systems of eigenvalues arising from classical automorphic forms (andtheir $-adic interpolations), but also systems of Hecke eigenvalues arising fromtorsion cohomology classes (and their $-adic interpolations). We will discuss thispoint further in (3.1.3) below.

It will be helpful to say a little more about how one can use (8) to analyzecohomology. To this end, we decompose the space of automorphic forms A(Kp

f )χ−1

as the direct sum

A(Kpf )χ−1 = Acusp(Kp

f )χ−1 ⊕AEis(Kpf )χ−1 (9)


of the cuspforms and the forms which are orthogonal to the cuspforms (in A(Kpf )χ)

under the Petersson inner product (i.e. the L2 inner product of functions onG(Q)\G(A)/A◦∞Kp

f )). We label this complement to the space of cuspforms withthe subscript Eis for Eisenstein, although its precise relationship with the space ofEisenstein series can be complicated; see e.g. [47], [39], and [40]. Note that if thequotients Y (Kf ) are compact, then A(Kp

f )χ−1 = Acusp(Kpf )χ−1 .

The decomposition (9) is a direct sum of G∞ × G-representations, and so thecohomology Hi

(g, k;Acusp(Kp

f )χ−1 ⊗ VC)

is a G-invariant direct summand of thecohomology Hi

(g, k;A(Kp

f )χ−1 ⊗ VC), which via the isomorphism (8) we identify

with a G-invariant direct summand Hicusp(VC) of Hi(VC). It is also TΣ-invariant.

We may choose an everywhere positive element of A(Kpf ) on which G(A) acts

via a character extending χ2, multiplication by which induces an isomorphism be-tween Acusp(Kp

f )χ−1 and Acusp(Kpf )χ. The Petersson (i.e. the L2) inner product

between Acusp(Kpf )χ−1 and Acusp(Kp

f )χ, taken together with this isomorphism,thus induces a pre-Hilbert space structure on Acusp(Kp

f )χ−1 , with respect to whichthe G∞ × G-action is unitary up to a character. In particular, we find thatAcusp(Kp

f )χ−1 is semisimple as a G∞ × G-representation, and so decomposes asa direct sum Acusp(Kp

f )χ−1 =⊕

π∞⊗πpπ∞ ⊗ πp ⊗M(π∞ ⊗ πp), where the direct

sum is taken over (a set of isomorphism class representatives of) all the irreducibleadmissible representations of G∞ ⊗ G over C on which A◦∞ acts via χ−1 (whichfactor as the tensor product π∞ ⊗ πp of an irreducible admissible representationπ∞ of G∞ on which A◦∞ acts via χ−1, and an irreducible admissible smooth repre-sentation πp of G), and M(π∞⊗πp) := HomG∞×G

(π∞⊗πp,Acusp(Kp

f )χ−1

)is the

(finite-dimensional) multiplicity space of π∞ ⊗ πp in Acusp(Kpf )χ−1 . Consequently,

we obtain a direct sum decomposition

Hicusp(VC) ∼= Hi

(g, k;Acusp(Kp

f )χ−1 ⊗ VC)

∼=⊕

π∞⊗πp

Hi(g, k;π∞ ⊗ VC)⊗ πp ⊗M(π∞ ⊗ πp) (10)

(the point here being that the (g, k)-cohomology depends only on the structure ofπ∞ ⊗ πp as a G∞-representation, which is to say, only on π∞).

In this way, we can speak of the contribution of each of the irreducible sum-mands π∞ ⊗ πp of Acusp(Kp

f )χ−1 to the cohomology Hicusp(VC). In the case when

Hi(g, k;π∞⊗VC) is non-zero, so that π∞⊗πp actually does contribute to cohomol-ogy, the multiplicity space M(π∞ ⊗ πp) is naturally a TΣ-module, and the directsum decomposition (10) is compatible with the actions of G and TΣ (where G-actson πp, and TΣ acts on M(π∞ ⊗ πp)).

We can now explain the significance of the quantities l0 and q0 introducedabove. Namely, q0 is the lowest degree in which tempered π∞ can admit non-trivial (g, k)-cohomology. Furthermore, among the tempered representations ofG∞, it is precisely the fundamental tempered representations (i.e. those which areinduced from discrete series representations of the Levi subgroup of a fundamentalparabolic subgroup of G∞) which can admit non-zero cohomology at all, and they

10 Matthew Emerton

do so precisely in degrees between q0 and q0 + l0 [10, Ch. III, Thm. 5.1]. (This is arange of degrees of length l0 + 1 symmetric about d/2 — which is one-half of thedimension of the quotients Y (Kf ).) A fundamental theorem of Harish-Chandrastates that, when G is semisimple, the group G∞ admits discrete series represen-tations precisely if l0 = 0; in this case q0 is equal to d/2, and the fundamentaltempered representations are precisely the discrete series representations. Key ex-amples for which l0 = 0 are given by groups G for which G∞ is compact, and (thesemisimple parts of) groups giving rise to Shimura varieties.

One representation of G∞ × G that always appears in A(Kpf ) is the trivial

representation 1. Thus there are induced morphisms

Hi(g, k;1)→ Hi(g, k;A(Kp

f )1) ∼= C⊗O Hi, (11)

whose sources are naturally identified with the cohomology spaces of the compactdual to the symmetric space G◦∞/A◦∞K◦

∞. If the Y (Kf ) are not compact, then 1 liesin AEis(K

pf )1, but it is typically not a direct summand, and so the morphisms (11)

are typically not injective. Nevertheless, when i is small they will be injective,and indeed (if e.g. G is split, semisimple, and simply connected) an isomorphism,and these isomorphisms play a key role in the theory of homological stability [9].We discuss the interaction of these isomorphisms with the theory of completedcohomology (in the case when G = SLN ) in (2.2.3) below.

2.1.7. Dimension theory. For any compact open subgroup Kp ⊂ G, the com-pleted group ring O[[Kp]] is (left and right) Noetherian, and of finite injectivedimension as a module over itself; more precisely, its injective dimension is equalto dim G+1. This allows us to consider a (derived) duality theory for finitely gen-erated O[[Kp]]-modules: to any such module M we associate the Ext-modulesEi(M) := Exti

O[[Kp]](M,O[[Kp]]), for i ≥ 0 (which necessarily vanish if i >dim G + 1); these are again naturally O[[Kp]]-modules. (We use the left O[[Kp]]-module structure on O[[Kp]] to compute the Ei, and then use the right O[[Kp]]-module structure on O[[Kp]], converted to a left module structure by the usualdevice of applying the anti-involution k 7→ k−1 on Kp, to give the Ei an O[[Kp]]-module structure.)

An important point is that Ei(M) is canonically independent of the choice of Kp

(in the sense that we may regard M as an O[[K ′p]]-module, for any open subgroup

K ′p of Kp, but Ei(M) is canonically independent of which choice of K ′

p we make todefine it). This has the consequence that if the Kp-representation on M extendsto a G-representation, then the Ei(M) are also naturally G-representations.

The Poincare duality spectral sequence (7) may thus be rewritten as

Ei,j2 = Ei(Hj) =⇒ HBM

d−(i+j).

All the objects appearing in it are G-representations, and it is G-equivariant.If Kp is p-torsion free (which will be true provided Kp is sufficiently small) then

the ring O[[Kp]] is in fact Auslander regular [55]. If Kp is furthermore pro-p, thenthe ring O[[Kp]] is an integral domain (in the sense that it contains no non-trivial


left or right zero divisors). This has various implications for the theory of finitelygenerated O[[Kp]]-modules. For example, it is reasonable to define such a moduleto be torsion if every element has a non-zero annihilator in O[[Kp]] (where Kp ischosen small enough to be pro-p and p-torsion free). More generally, we may makethe following definition.

Definition 2.3. If M is a finitely generated O[[Kp]]-module, then the codimensioncd(M) of M is defined to be the least value of i for which Ei(M) 6= 0; the dimensiondim M of M is defined to be dim G + 1− cd(M).

If G is a torus, so that Kp is commutative, then dim M is precisely the dimen-sion of the support of the localization of M over SpecO[[Kp]]. In general O[[Kp]]is non-commutative, and hence doesn’t have a Spec over which we can localizea finitely generated module M . Nevertheless, the quantity dim M behaves as ifit were the dimension of the support of M in the (non-existent) Spec of O[[Kp]].(See e.g. [55, Prop. 3.5].) As one example, we note that M is torsion (in the sensedefined above) if and only if cd(M) ≥ 1. As another, we note that cdEi(M) ≥ i,with equality when i = cdM . Also, we note that cdM = ∞ (so, morally, thesupport of M is empty) precisely when M = 0.

By Theorem 2.2, completed homology is finitely generated over O[[Kp]], and sothese dimension-theoretic notions apply to it. Information about the (co)dimensionof Hi can be obtained by analyzing the Poincare duality spectral sequence, since thefunctors Ei appear explicitly in it, and there is a tension in this spectral sequencebetween the top degree for the duality of O[[Kp]]-modules (which is dim G + 1)and the top degree for usual Poincare duality (which is d), which can sometimes beexploited. (See e.g. (2.2.2) below.) In making any such analysis, it helps to havesome a priori information about the (co)dimensions of the Hi. This is providedby the following result.

Theorem 2.4. [23] Suppose that G is semisimple. Then Hi is torsion (i.e. itscodimension is positive) unless l0 = 0 and i = q0, in which case cd Hi = 0.

We make a much more precise conjecture about the codimensions of the Hi inConjecture 3.2 below.

2.2. Examples. We illustrate the preceding discussion with some examples.

2.2.1. GL2 of Q. If G = GL2, then the quotients Y (Kf ) are classical modularcurves. In particular, they have non-vanishing cohomology only in degrees 0 and 1,and so we consider completed cohomology in degrees 0 and 1. Since the cohomologyof a curve is torsion free, we have H0 = H0 ∼= C

(∆×Z×p ,O

), the space of continuous

O-valued functions on the product ∆×Z×p , where ∆ is a finite group that dependson the choice of tame level, and so H0 (which is simply the O-dual of H0) isisomorphic to O[[∆ × Z×p ]], which is of dimension two as a module over O[[Kp]](for any compact open Kp ⊂ GL2(Qp)). We also have H1 = H1. Again, H1 is theO-dual of H1, and Theorem 2.4 shows that cd H1 = 0. (Strictly speaking, we have

12 Matthew Emerton

to apply the theorem to SL2 rather than GL2, but it is then easy to deduce thecorresponding result for GL2, by considering the cup-product action of H0 on H1.)

2.2.2. SL2 of an imaginary quadratic field. Suppose that G = ResF/Q SL2,where F is an imaginary quadratic extension of Q. The quotients Y (Kf ) are thenconnected, non-compact three-manifolds. The relevant (co)homological degreesare thus i = 0, 1, and 2. Since the Y (Kf ) are connected we see that H0 = H0 = O.Theorem 2.4 implies that H1 has positive codimension. A consideration of thePoincare duality spectral sequence then shows that H2 = 0, and that H1 is ofcodimension 1. This computation exploits both the fact that H0 6= 0, and the gapbetween 3 (the dimension of the Y (Kf )) and 6 (the dimension of G).

Since H1 with coefficients in O of any space is $-torsion free, we see that H1 is$ torsion-free, and coincides with the O-dual of H1, while H2 is $-torsion, and isnaturally identified with the Pontrjagin dual of the O-torsion submodule of H1 [24,Thm. 1.1]. We conjecture that in fact H1 is $-torsion free (see Conjecture 3.2),and thus that H2 = 0.

2.2.3. SLN of Q in low degrees. We first discuss H0 and H1, before turningto a discussion of higher degree cohomology from the point of view of homologicalstability.

The quotients Y (Kf ) are connected, and so H0 = O. If N ≥ 3, then SLN

satisfies the congruence subgroup property. Furthermore, the groups SLN (Z`)are perfect for all values of `. Combining these two facts, we see that if Γ(pr)denotes the usual congruence subgroup of level pr (i.e. the kernel of the surjectionSLN (Z)→ SLN (Z/pr)), then H1

(Γ(pr),O) = O⊗Zp Γ(pr)ab ∼= O⊗Zp Γ(pr)/Γ(p2r).

Thus, if we take the tame level to be trivial (i.e. “level one”), then we see that thetransition maps in the projective system defining H1 are eventually zero, implyingthat H1 = 0. Similarly, we see that H1 = H1 = 0. (If we allowed a more generaltame level, then H1 could be non-zero, but would be finite. Since H1 and H1 arealways torsion free, they would continue to vanish.)

We now consider cohomology in higher degree, but in the stable range, in thesense that we now explain. (The tame level can now be taken to be arbitrary.)Borel’s result [9] on homological stability for SLN shows that when i is sufficientlysmall (compared to N), the cohomology L ⊗O Hi is independent of N ; indeed,it consists precisely of the contributions to cohomology arising from the trivialautomorphic representation, in the sense discussed in (2.1.6). If we define Hi

stab

to be the stable value of Hi, then Borel shows that⊕

Histab (as an algebra under

cup product) is isomorphic to an exterior algebra generated by a single element ineach degree i = 0, 5, 9, 13, . . . .

In fact, stability also holds for completed cohomology.

Theorem 2.5. [25] If i is sufficiently small compared to N , then Hi is independentof N , is finitely generated as an O-module, and affords the trivial representationof G.

We write Histab to denote the stable value of Hi. F. Calegari has succeeded


in computing Histab modulo torsion, contingent on a natural non-vanishing conjec-

ture for certain special values of the p-adic ζ-function (a conjecture which holdsautomatically if p is a regular prime).

Theorem 2.6. [22, Thm. 2.3] Suppose either that p is a regular prime, or thatappropriate special values of the p-adic zeta function are non-zero. Then thereis an isomorphism of graded vector spaces

⊕i≥0 L ⊗O Hi

stab∼= L[x2, x6, x10, . . .]

(where L[x2, x6, x10, . . .] denotes the graded ring generated by the indicated sequenceof polynomial variables xi, i ≡ 2 mod 4, with xi placed in degree i).

Note in particular that⊕

i L ⊗O Histab vanishes in all odd degrees, while⊕

i L ⊗O Histab is generated by classes in odd degrees. Thus, when G = SLN ,

the map (3) is identically zero (modulo torsion) when i lies in the stable range!Assuming the hypothesis of the theorem, we conclude that the Borel classes

(i.e. the non-zero elements of L ⊗O Histab) become infinitely divisible when we

pull them back up the p-power level tower. It is interesting to consider how theyreappear in the Hochschild–Serre spectral sequence. If we again ignore torsion,then Hi

(SLN (Zp), L

)coincides with the Lie algebra cohomology of slN [48], which

is (stably in N) an exterior algebra on generators in degrees 3, 5, 7, . . . . Sinceall of the non-zero Hi

stab have trivial G-action, we see that we obtain non-trivialExt terms in odd degrees in the Hochschild–Serre spectral sequence, and the Borelclasses are recovered from these.

If we consider the short exact sequence (6) for non-zero even degrees, we see(continuing to assume the hypothesis of the theorem) that Hi vanishes (modulotorsion), while Hi is non-zero. Thus the term TpH

i+1 must be non-zero; thisprovides a rather compelling illustration of the manner in which torsion classescan accumulate as we pass to the limit of the p-power level tower.

2.2.4. Groups that are compact at infinity. If G∞ (or, more generally, thequotient G∞/A∞) is compact, then A◦∞K◦

∞ equals G◦∞, and so the quotientsY (Kf ) are simply finite sets. Thus the only interesting degree of cohomologyis i = 0. In this case the inverse limit Y (Kp

f ) := lim←−KpY (KpK

pf ) is a pro-finite

set, which is in fact a compact p-adic analytic manifold, equipped with an analyticaction of G. If Kp is taken sufficiently small (small enough that the G(Q)-actionon G(A)/G◦∞KpK

pf is fixed-point free), then Kp acts with trivial stabilizers on

Y (KpKpf ), and Y (KpK

pf ) is the disjoint union of finitely many (say n) principal

homogeneous spaces over Kp.One immediately verifies that H0 ∼= C

(Y (Kp

f ),O), the space of continuous

O-valued functions on Y (Kpf ), while H0

∼= O[[Y (Kpf )]], the space of O-valued

measures on Y (Kpf ) (which is the O-dual of C

(Y (Kp

f ),O)). In particular, if Kp is

sufficiently small, then we see that H0 is free of rank n over O[[Kp]].It is natural (e.g. in light of Gross’s definition of algebraic modular forms in this

context [42]) to define C(Y (Kp

f ),O)

to be the space of p-adic automorphic formson G(A) of tame level Kp

f , and so in this case we see that completed cohomologydoes indeed coincide with the natural notion of p-adic automorphic forms.

14 Matthew Emerton

3. p-adic Langlands

We describe the manner in which we expect completed cohomology, and the Heckealgebra acting on it, to be related to deformation rings of global Galois represen-tations. This conjectural relationship suggests various further conjectures, as weexplain, as well as a relationship to a hypothetical p-adic local Langlands corre-spondence.

3.1. The connection between completed cohomology and Ga-lois representations. If π is an automorphic representation of G(A) forwhich Hi(g, k;π∞ ⊗ VC) 6= 0 for some algebraic representation V of G and somedegree i, then π is C-algebraic [21, Lemma 7.2.2]. Thus, we expect that associatedto π there should be a continuous representation of GQ into the Qp-valued points ofthe C-group of G [21, Conj. 5.3.4]. In light of the isomorphism (8), we thus expectthat the systems of Hecke eigenvalues that occur in Hi(V) should have associatedrepresentations of GQ into the C-group of G. For systems of Hecke eigenvaluesoccurring on torsion classes in cohomology, conjectures of Ash [2] again suggestthat there should be associated Galois representations.

These expectations have been proved correct in many cases; for example if G isthe restriction of scalars to Q of GLn over a totally real or a CM number field. (See[45] in the case of characteristic zero systems of eigenvalues, and [53] in the caseof torsion systems of eigenvalues. See also [43] for an overview of these results.)

Returning for a moment to the general case, one further expects that the Galoisrepresentations obtained should be odd, in the sense that complex conjugation inGQ maps to a certain prescribed conjugacy class in the C-group of G. (See [7,Prop. 6.1] for a description of the analogous conjugacy class in the L-group of G.)

Let m denote a maximal ideal in TΣ, write F := TΣ/m, and let F denote a chosenalgebraic closure of F. Then, by the above discussion, we believe that associatedto m there should be a continuous representation of GQ into the F-valued pointsof the C-group of G. Assuming that this representation exists, we will denote itby ρm.

To be a little more precise: the representation ρm should have the property thatit is unramified at the primes outside Σ, and that for any ` 6∈ Σ, the semisimplepart of ρm(Frob`) should be associated to the local-at-` part of the system ofHecke eigenvalues by a suitable form of the Satake isomorphism (see [21] and [42]).In general, this condition may not serve to determine ρm uniquely; even in thecase G = GLn, it determines ρm only up to semisimplification; for other choicesof G, in addition to this issue, one can have the additional phenomenon that evena representation which doesn’t factor through any parabolic subgroup of the C-group is not determined by its pointwise conjugacy classes (“local conjugacy doesnot determine global conjugacy”). We do not attempt to deal with this issue ingeneral here; rather we simply ignore it, and continue our discussion as if ρm wereunambiguously defined.

As we observed above, we may regard Tm (or its Spec) as interpolating systemsof Hecke eigenvalues associated to classical C-algebraic automorphic forms, and/orto torsion classes in cohomology. Since the deformations of ρm can be interpolated


into a formal deformation space, it is then reasonable to imagine that we maydeform ρm to a representation of GQ into the Tm-valued points of the C-groupof G. Here again there are many caveats: firstly, if ρm is “reducible” (i.e. factorsthrough a proper parabolic of the C-group), then we would have to work with someform of pseudo-character or determinant, as in [28] and [53]; also, there is a questionof rationality, or “Schur index” — it may be that if we want our deformation ofρm to be defined over F, then we may have to extend our scalars, or else replacethe C-group by an inner twist. Again, we don’t attempt to address these issueshere.

Rather, we begin with some general conjectures about dimension and vanishingthat are motivated by the preceding discussion, and then continue by discussingthe relationships between completed cohomology and p-adic Hodge theory and apossible p-adic local Langlands correspondence. Finally, we turn to a discussion ofsome specific examples, where we can make our generalities more precise.

3.1.1. Conjectures on dimension and vanishing. We begin by making asomewhat vague conjecture, which can be thought of as a rough expression of ourhopes for reciprocity in the context of global p-adic Langlands: namely (continuingwith the notation introduced in the preceding discussion), we conjecture that Tm

is universal (in some suitable sense) for parameterizing odd formal deformations ofρm whose ramification away from p is compatible with the tame level structure Kp

f

(via some appropriate form of $-adic local Langlands for the group G at primes` - p which, again, we won’t attempt to formulate here; but see [36] in the casewhen G = GLn). The global Euler characteristic formula for Galois cohomologylets us compute the expected dimension of such a universal deformation ring, andthis motivates in large part the following concrete conjecture [24, §8].

Conjecture 3.1. Each local factor Tm of TΣ is Noetherian, reduced, and $-torsion free, of Krull dimension equal to dim B + 1 − l0, where dim B denotes thedimension of a Borel subgroup B of G.

This is known in some cases (which we will recall below). We know of no way toprove the Noetherianness of Tm, or the statement about its Krull dimension, otherthan to relate the Hecke algebra to a deformation ring of Galois representations,and then use techniques from the theory of Galois representations to compute thedimension.

In some situations we can prove that Tm is torsion free and reduced (see e.g. thediscussion of (3.1.2) below), and it seems reasonable to conjecture these propertiesin general. Indeed, these properties are closely related to the following conjecture[24, Conj. 1.5].

Conjecture 3.2. Hi = 0 if i > q0, while cd Hq0 = l0, and cd Hi > l0 + q0 − i ifi < q0. Furthermore, Hq0 is $-torsion free, and TΣ acts faithfully on Hq0 .

As noted in [24, Thm. 1.6], the truth of this conjecture for G and all its Levisubgroups implies the analogous statement for HBM

i . Also, the vanishing conjecturefor G×G implies (via the Kunneth formula) the torsion-freeness of Hq0 .

16 Matthew Emerton

One of the ideas behind this conjecture is that “all the interesting Hecke eigen-values should appear in degree q0”. This is inspired by the fact (recalled in (2.1.6))that tempered automorphic representations don’t contribute to cohomology in de-grees below q0, so that the Galois representations associated to the systems ofHecke eigenvalues appearing in degrees below q0 should be “reducible” (i.e. fac-tor through a proper parabolic subgroup of the C-group). In a Galois deformationspace with unrestricted ramification at p, the reducible representations should forma proper closed subset, and so should be approximable by “irreducible” Galois rep-resentations (i.e. representations which don’t factor through a proper parabolic).Thus we don’t expect to see any systems of Hecke eigenvalues in degrees below q0

which can’t already be observed in degree q0, and hence we expect that TΣ willact faithfully on Hq0 .

We also expect that Hi should be “small” if i < q0, since the possible systemsof Hecke eigenvalues which it can carry are (or should be) constrained. However,if the Hi are small enough for i < q0, then the Poincare duality spectral sequence(combined with an analysis of the completed cohomology of the boundary, whichcan be treated inductively, by reducing to the case of lower rank groups) impliesthe vanishing of Hi in degrees > q0.

If we believe that TΣ acts faithfully on Hq0 , and that TΣ has dimensiondim B + 1 − l0 (as predicted by Conjecture 3.1), then conjecturing that Hq0 hascodimension l0 is morally equivalent to conjecturing that the fibres of Hq0 over thepoints of Spec TΣ are of dimension dim G/B. This latter statement fits nicely withthe fact that generic irreducible representations of G := G(Qp) have Gelfand–Kirillov dimension equal to dim G/B, and the analogy between the dimensionof O[[Kp]]-modules and Gelfand–Kirillov dimension [24, Remark 1.19]. Unfor-tunately, we don’t know how to make this idea precise, since we don’t know howto prove (in any generality) this relationship between the Krull dimension of TΣ,the dimension of Hq0 , and the dimension of the fibres of the latter over points ofSpec TΣ, even assuming that TΣ acts faithfully on Hq0 . Nevertheless, the idea thatthese dimensions should be related is an important motivation for the conjecture.

We note that if TΣ acts faithfully on Hq0 , and Hq0 is $-torsion free, then TΣ

is $-torsion free. This motivates the conjecture of $-torsion freeness in Conjec-ture 3.1.

Conjecture 3.2 has essentially been proved by P. Scholze in many cases for whichthe group G gives rise to Shimura varieties [53, Cor. IV.2.3]. (He has non-strictinequalities rather than strict inequalities for the codimension of the cohomologyin degrees < q0.)

We mention one more example here, namely the case when G is the restrictionof scalars of SL2 from an imaginary quadratic field. In this case we have l0 = 1,and the codimension statement of Conjecture 3.2 follows from the computationsketched in (2.2.2).

3.1.2. Localization at a non-Eisenstein system of Hecke eigenvalues.Suppose that m is a maximal ideal in TΣ, and that the associated representa-


tion ρm (which we assume exists) is “irreducible”, i.e. does not factor through anyproper parabolic subgroup of the C-group of G. We refer to such a m as non-Eisenstein. For any TΣ-module M , we write Mm := Tm ⊗TΣ M to denote thelocalization of M at m.

As already noted, we expect that all the systems of Hecke eigenvalues appear-ing in cohomological degrees < i are “reducible”, i.e. do factor through a properparabolic subgroup, and so we conjecture that, when m is a non-Eisenstein maxi-mal ideal, Hi

(Y (KpK

pf ),W)m = 0 for all i < q0 and all local systemsW associated

to Kp-representations W on finitely generated O-modules (where Kp is any suffi-ciently small compact open subgroup of G).

We remark that if we ignore torsion, then this follows (at least morally) fromArthur’s conjectures [1]. (Namely, the result is true for the boundary cohomology,and hence it suffices to check it for the interior cohomology, i.e. the image ofcompactly supported cohomology in the usual cohomology. However, the interiorcohomology is contained in the L2-cohomology, and now Arthur’s conjectures implythat any automorphic representation π contributing to L2-cohomology that is non-tempered at ∞ — as must be the case for an automorphic representation thatcontributes to cohomology in degree < q0 — is non-tempered at every prime, andhence should give rise to a reducible Galois representation.) We believe that thisstatement should be true for torsion cohomology as well.

Let us suppose, then, that Hi(Y (KpK

pf ),W)m = 0 for all i < q0. Just con-

sidering the case when W = O (the trivial representation), we find that Him = 0

for i < q0. Certainly it should be the case that Hi(∂(KpK

pf ),W)m = 0 when m

is non-Eisenstein. We then find that Hic

(Y (KpK

pf ),W)m = 0 for all i < q0 as

well. Suppose now that l0 = 0, so that q0 = d/2. Classical Poincare duality thengives that Hi

(Y (KpK

pf ),W)m = 0 for all i > q0. Presuming that Conjecture 3.2

is true, so that Hi = 0 for i > q0 and Hq0 is $-torsion free, a consideration ofthe Hochschild–Serre spectral sequence shows that Exti

O[[Kp]](W∨, Hq0

m ) = 0 for alli > 0 and all representations of Kp on finitely generated torsion free O-modules.From this one easily deduces that (Hq0)m is projective as an O[[Kp]]-module.

In short, we have given a plausibility argument for the following conjecture,which refines Conjecture 3.2 in the context of localizing at a non-Eisenstein maxi-mal ideal.

Conjecture 3.3. If m is a non-Eisenstein maximal ideal in TΣ, and if l0 = 0,then (Hi)m = 0 for i 6= q0, and (Hq0)m is a projective O[[Kp]]-module, for anysufficiently small subgroup Kp of G. (Here sufficiently small means that G(Q) actswith trivial stabilizers on G(A)/A◦∞K◦

∞KpKpf .)

If l0 = 0, and if Conjecture 3.3 holds for some non-Eisenstein maximal ideal m,then we see (from the Hochschild–Serre spectral sequence) that for any sufficientlysmall Kp (as in the statement of the conjecture) and any representation of Kp ona finitely generated torsion free O-module W , we have Hi

(Y (KpK

pf ),W)m = 0 if

i 6= q0, while Hq0(Y (KpK

pf ),W)m

∼= HomO[[Kp]](W∨, Hq0m ). In particular, we see

that Hq0(Y (KpK

pf ),W)m is $-torsion free. We also see that Hq0

m = Hq0m (since

18 Matthew Emerton

Hq0+1m vanishes, and hence so does TpH

q0+1m ), and that Tm acts faithfully on Hq0

m .Presuming that Hi

(∂(KpK

pf ),W

)m

= 0 (which should certainly be true whenm is non-Eisenstein), we conclude that the natural map

Hq0c

(Y (KpK

pf ),W)m → Hq0

(Y (KpK

pf ),W)m

is an isomorphism, and thus that Hq0(Y (KpK

pf ),W)m consists entirely of interior

cohomology. In particular (being torsion free) it embeds into the L2-cohomology.From this we deduce that the image of TΣ acting on Hq0

(Y (KpK

pf ),W)m is re-

duced, and hence (considering all possible Kp and W ) that Tm itself is reduced,as well as being $-torsion free. This provides some evidence for the reducednessand torsion-freeness statements in Conjecture 3.1.

We now recall some known results in the direction of Conjecture 3.3. In thecase when G = GL2, the concept of a non-Eisenstein maximal ideal is well-defined,and the above conjecture holds [32, Cor. 5.3.19]. In the case when G = GLN ,this notion is again well-defined, and the vanishing of (Hi)m is proved for i in thestable range in [25]. Again, if G is a form of U(n− 1, 1) over Q, then the notion ofa non-Eisenstein maximal ideal is well-defined, and in [34] we prove vanishing of(Hi)m for i in a range of low degrees, for certain maximal ideals m. In particular, inthe case of U(2, 1), we are able to deduce Conjecture 3.3, provided not only that mis non-Eisenstein, but that the associated Galois representation ρm has sufficientlylarge image, and is irreducible locally at p, satisfying a certain regularity condition.

As one more example, note that if G∞/A∞ is compact (in which case certainlyl0 = 0), then Conjecture 3.3 holds. Indeed, in this case we saw in (2.2.4), forsufficiently small Kp, that H0 is free over O[[Kp]], even without localizing at anon-Eisenstein maximal ideal.

If l0 6= 0, then we don’t expect that (Hq0)m should be projective over O[[Kp]];indeed, this would be incompatible with Conjecture 3.2. Rather, we expect thatit should be pure of codimension l0, in the sense of [55, Def. 3.1].

3.1.3. The relationship with p-adic Hodge theory. Let us continue to sup-pose that l0 = 0 and that m is a non-Eisenstein maximal ideal in TΣ, and let ussuppose that Conjecture 3.3 holds. As we have observed, this implies that for any(sufficiently small) compact open subgroup Kp of G, and any representation W ofKp on a finitely generated torsion free O-module, we have

Hi(Y (KpK

pf ),W)m

∼= HomO[[Kp]](W∨, Hq0m ) ∼= HomO[[Kp]]

((Hq0)m,W

)(the first isomorphism being given by the Hochschild–Serre spectral sequence, andthe second by duality).

We now suppose that W is an O-lattice in an irreducible algebraic represen-tation V of G over L. Since Hi

(Y (KpK

pf ),W)m is $-torsion free, it is a lattice

in Hi(Y (KpK

pf ),V)m. Considering the description of this cohomology in terms

of automorphic forms (as in (2.1.6)) and the conjectures regarding Galois repre-sentations associated to automorphic forms [21, Conj. 5.3.4], we infer that theGalois representations associated to the systems of Hecke eigenvalues appearing in


Hi(Y (KpK

pf ),V)m should be potentially semistable locally at p, with Hodge–Tate

weights related to the highest weight of the algebraic representation V .Since m is non-Eisenstein, it should make sense to speak of the formal defor-

mation ring Rρmof ρm, and we conjecture that there is a natural isomorphism

Tm∼= Rρm

. We see that the Tm-modules Hi(Y (KpK

pf ),W)m (where W ranges

over the various lattices in the various algebraic representations V ) should then besupported on the set of points of Spec Rρm

corresponding to deformations of ρm

that are potentially semistable at p.The Fontaine–Mazur conjecture [38], when combined with Langlands reci-

procity, should give a precise description of the support of Hi(Y (KpK

pf ),W)m,

in purely Galois-theoretic terms. Namely, any potentially semistable deformationof ρm of the appropriate Hodge–Tate weights should be motivic, and hence shouldbe associated to an automorphic form on the quasi-split inner form of G. Whetherthis automorphic form can then be transferred to G, and can contribute to coho-mology at level KpK

pf , should be answered by an analysis of the local Langlands

correspondence for G.Conversely, if we can prove directly that the support of Hi

(Y (KpK

pf ),W)m is

as predicted by the Fontaine–Mazur–Langlands conjecture, then we can deducethis conjecture for deformations of ρm that are classified by Spec Tm. We recallin (3.2.2) below how this strategy is compatible with an optimistic view-point onhow p-adic local Langlands might behave. We briefly recall in (3.3.1) how thisstrategy was employed in [32] and [46] in the case when G = GL2.

Let us fix a sufficiently small open subgroup Kp. Since (Hq0)m is projective overO[[Kp]] by assumption, it is a direct summand of a finitely generated free O[[Kp]]-module, and hence Hq0

m is a direct summand of C(Kp,O)L

n, for some n > 0. ThusL ⊗O Hq0

m is a direct summand of C(Kp, L)⊕n, the space of continuous L-valuedfunctions on Kp. Now the theory of Mahler expansions shows that the affine ringL[G] embeds with dense image in C(Kp, L) [51, Lemma A.1]. Recall that, as aG-representation, we have L[G] ∼=

⊕V V ⊗L HomG(V,L[G]), where V runs over

(a set of isomorphism class representatives of) all irreducible representations of G.(At this point, we assume for simplicity that L is chosen so that all the irreduciblerepresentations of G over Qp are in fact defined over L; thus these irreducible Vare in fact absolutely irreducible.) Now the inclusion of L[G] into C(Kp, L) inducesan isomorphism HomG(V,L[G]) ∼= HomKp

(V, C(Kp, L)

)(both are naturally iden-

tified with V ∨, the contragredient of V ). We conclude that the natural morphism⊕V V ⊗L HomKp

(V, C(Kp, L)

)→ C(Kp, L) is injective with dense image. This

property is clearly preserved under passing to direct sums and direct summands,and hence the natural morphism

⊕V V ⊗LHomKp

(V, Hq0m ⊗OL)→ Hq0

m ⊗OL is alsoinjective with dense image. Replacing V by V ∨ (which clearly changes nothing,since we are summing over all irreducible algebraic representations of G), and re-calling that (by Hochschild–Serre) HomKp(V ∨, Hq0

m ⊗O L) ∼= Hq0(Y (KpK

pf ),V

)m

,

we find that Tm acts faithfully on⊕

V Hq0(Y (KpK

pf ),V

)m

(since, as we notedabove, it follows from Conjecture 3.3 that Tm acts faithfully on Hq0

m ).Assuming that we can identify Tm with a deformation space of Galois repre-

20 Matthew Emerton

sentations, and that we know that Hq0(Y (KpK

pf ),V

)m

is indeed supported on anappropriate locus of potentially semistable deformations, the preceding analysisshows that these loci (as V varies) are Zariski dense in Spec Tm.

In forthcoming work [37], the author and V. Paskunas will apply a more sophis-ticated version of this argument to deduce additional Zariski density statementsfor various collections of potentially semistable loci in global Galois deformationspaces.

Note that these density results rely crucially on the projectivity statement ofConjecture 3.3 for their proof. As we already noted, we can’t expect such a state-ment to be true when l0 > 0, and (at least if G is semisimple, or, more generally, ifthe semisimple part of G satisfies l0 > 0) we don’t expect the potentially semistablepoints to be Zariski dense in global deformation spaces in this case. (See [26] foran elaboration on this point.) In particular, in such contexts, we don’t expect thatHq0 will equal Hq0 , and so Hq0 , and Spec TΣ, should receive a non-trivial contri-bution from torsion classes (via the term TpH

q0+1 in the exact sequence (6)).

3.2. p-adic local Langlands. Until now, our discussion has been entirelyglobal in nature. We now turn to describing how these global considerations mightbe related to a possible p-adic local Langlands correspondence.

3.2.1. The basic idea. Let us return for a moment to the direct sum decompo-sition (10). Local-global compatibility for classical Langlands reciprocity says thatif π∞⊗πp is a direct summand of Acusp(Kp

f )χ−1 which contributes to cohomology,lying in a Hecke eigenspace that corresponds to some p-adic Galois representa-tion ρ, then the local factor πp and the Weil–Deligne representation attached to ρ(which should be defined, since ρ should be potentially semistable at p) shouldcorrespond via the local Langlands correspondence (or perhaps more generally viathe local form of Arthur’s conjectures [1], if π∞ and πp are not tempered). Inparticular, the local factor at p and the local Galois representation at p should berelated in a purely local manner.

The basic idea of a p-adic local Langlands correspondence is that the sameshould be true when we take into account the structure of completed cohomology.To explain this, we continue the discussion of the preceding paragraph, and, inaddition, we place ourselves in the context introduced in (3.1.3) (in particular, wecontinue to assume that the hypotheses and conclusions of Conjecture 3.3 hold).Thus, we assume that ρ is a deformation of ρm, and that π∞ and πp are tempered,so that we have an embedding πp ↪→ Hq0(V)m

∼= lim−→KpHomKp(V ∨, L⊗O Hq0

m ). We

may rewrite this as an embedding V ∨⊗L πp ↪→ L⊗O Hq0m and we can then take the

closure of V ∨ ⊗ πp in Hq0(V)m, to obtain a unitary Banach space representationV ∨ ⊗L πp of G over L. A slightly more refined procedure is to form the intersection

(V ∨⊗L πp)◦ := (V ∨⊗L πp)∩ Hq0m (the intersection being taken in L⊗O Hq0

m ). Thisis a G-invariant O-lattice contained in V ∨⊗L πp, and the Banach space V ∨ ⊗L πp

is then obtained by completing V ∨ ⊗L πp with respect to this lattice. A naturalquestion to ask, then, in the spirit of a local Langlands correspondence and local-


global compatibility, is whether V ∨ ⊗L πp depends only on the restriction to p ofthe associated Galois representation ρ. Since by assumption m is non-Eisenstein,the Galois representation ρ should admit an essentially unique integral model ρ◦,and we could further ask whether (V ∨ ⊗L πp)◦ depends only on the restriction top of ρ◦. This question goes back to C. Breuil’s first work on the p-adic Langlandscorrespondence in the case when G = GL2 (see especially the introduction of [13]).It has been largely resolved in that case, but remains open in general.

3.2.2. An optimistic scenario. The most optimistic conjecture that one mightentertain regarding a p-adic local Langlands correspondence is that for any formaldeformation ring Rr parameterizing the deformations of a representation r of thelocal Galois group GQp into the F-valued points of the C-group of G, there is aprofinite Rr-module M , finitely generated over Rr[[Kp]] for some (and hence every)compact open subgroup Kp ⊂ G, equipped with a continuous G-action extendingits Kp-module structure, which realizes the p-adic local Langlands correspondencefor the group G in the following sense: in the context of (3.2.1) (and continuingwith the notation of that discussion), the fibre over the restriction to p of ρ◦ (whichis a point in Spec Rr, if r is the restriction of ρm to GQp) is isomorphic to the O-dualof (V ∨ ⊗L πp)◦. (Here we suppress the issue of whether Rr should be understoodto be a framed deformation ring, or a pseudo-deformation ring, or . . . .) In short,M should be a local analogue of the completed homology space (Hq0)m.

Since M is finitely generated over Rr[[Kp]], for any representation W of Kp on afinitely generated torsion free O-module, the Rr-module Homcont

Kp(M,W )d (where

d denotes the continuous O-dual) is a finitely generated Rr-module. Anotherproperty one might require of M is that when W is a Kp-invariant O-lattice inan irreducible algebraic G-representation V , then Homcont

Kp(M,W )d is supported

on an appropriate locus of potentially semistable representations, corresponding tothe fact that HomO[[Kp]]

((Hq0)m,W

)(which is isomorphic to Hq0

(Y (KpK

pf ),W

))

is supported on a locus of potentially semistable representations in Spec Tm.Ideally, one might ask for the support of Homcont

Kp(M,W )d to be the full poten-

tially semistable locus of appropriate Hodge–Tate weights (corresponding to thehighest weight of V ) and Weil–Deligne representations (corresponding to the par-ticular choice of Kp and the nature of the local Langlands correspondence for G).As we indicated in (3.1.3), such a result, combined with the local-global compati-bility between M and (Hq0)m, would prove that any global Galois representationcorresponding to a point of Spec Tm which is potentially semistable of appropriateHodge–Tate weights and with an appropriate Weil–Deligne representation, in factarises from a system of Hecke eigenvalues occurring in Hq0

(Y (KpK

pf ),V

)m

, thusverifying the Fontaine–Mazur–Langlands conjecture for such points.

Note that a local Galois representation lies in the support of HomcontKp

(M,W )d

precisely if the dual to the fibre of M at this point receives a non-zero Kp-equivariant homomorphism from W . In particular, the dual to the fibre at such apoint contains locally algebraic vectors. In the case of GL2, the idea of describingp-adic local Langlands for potentially semistable representations in terms of locallyalgebraic vectors goes back to C. Breuil’s first work on the subject [11, 12].

22 Matthew Emerton

The optimistic scenario described here has been realized for the group G =GL2(Qp); this is the theory of p-adic local Langlands for GL2(Qp) [6, 29, 50, 30](see also [5, 14]). Whether it can be realized for other groups, or is overly optimistic,remains to be seen.

3.3. Examples. We again illustrate our discussion with some examples.

3.3.1. GL2 of Q. As already mentioned, the theory of p-adic local Langlands forGL2(Qp) provides a structure satisfying all the desiderata of (3.2.2). The local-global compatibility between this structure and H1

m (the localization of completedcohomology at a non-Eisenstein maximal ideal of the Hecke algebra) has beenproved in [32] (see also [14, 15]) (under a mild hypothesis on the local behaviourof ρm). In particular, the strategy of (3.1.3) then applies to prove the Fontaine–Mazur–Langlands conjecture for points of Spec Tm.

Under the slightly stronger hypotheses that p is odd and ρm remains irreducibleon restriction to GQ(ζp) (the Taylor–Wiles condition) one can prove that Tm

∼= Rρm

[8], [32, Thm. 1.2.3]. Since Conjecture 3.3 holds in this context [32, Cor. 5.3.19], themethod of (3.1.3) (applied with Kp = GL2(Zp)) allows us to deduce the density ofthe crystalline loci in Spec Rρm

; the extension of this method to be described in [37]will allow us to deduce other density results about various potentially semistableloci. Here is one such simple variant: instead of considering the restriction toGL2(Zp) of the family V of algebraic representations of GL2, we instead considerthe family of representations σ ⊗ V , where σ is some fixed supercuspidal typeand V is algebraic. This allows us to show that potentially crystalline points of(any fixed) supercuspidal type are Zariski dense in Tm (and hence in Rρm

, if theTaylor–Wiles condition is satisfied).

3.3.2. A definite quaternion algebra ramified at p. Let D be the quater-nion algebra over Q that is ramified at ∞ and p, and let G = D×. Then G∞/A∞is compact, and so, as noted above, Conjecture 3.3 holds. The classical Jacquet–Langlands correspondence allows us to attach odd two-dimensional Galois repre-sentations to systems of Hecke eigenvalues. In particular, if m is non-Eisenstein,then Tm is a quotient of Rρm

.Assuming that p is odd and ρm satisfies the Taylor–Wiles condition, we con-

clude that in fact Tm∼= Rρm

. Indeed, we saw in the preceding example that thepotentially crystalline points of supercuspidal type are Zariski dense in Rρm

. SinceFontaine–Mazur–Langlands holds in this context, they all arise from classical mod-ular forms, and hence (by Jacquet–Langlands) from classical automorphic forms onD×. Thus Spec Tm contains a Zariski dense set of points in Spec Rρm

, and so thetwo Specs coincide. As one interesting consequence of this, we note that (assumingthat the Taylor–Wiles condition holds) the Hecke algebras at m for GL2 and D×

are naturally isomorphic (both being isomorphic to Rρm). One can think of this

as a p-adic interpolation of the classical Jacquet–Langlands correspondence. It isinteresting to note that even though under the classical Jacquet–Langlands corre-spondence automorphic forms on GL2 that are principal series at p don’t match


with a corresponding Hecke eigenform on D×, they are not excluded from thisp-adic Jacquet–Langlands correspondence.

As another consequence, note that if we choose Kp to be the units in themaximal order of (D⊗Qp)×, and apply the density argument of (3.1.3), we deducethat the representations which are genuinely semistable at p (i.e. semistable, andnot crystalline) are Zariski dense in Spec Tm, and hence in Spec Rρm

, provided theTaylor–Wiles condition holds.

3.3.3. Definite quaternion algebras over totally real fields. Consider thecase where G is the restriction of scalars to Q of the units D× in a totally definitequaternion algebra D over a totally real field F ; and suppose that F is unram-ified at p, and D is split at every prime in F above p. Put ourselves in thesituation of (3.2.1), with the additional assumption that V is the trivial represen-tation, and πp is a tamely ramified principal series. In this case πp contains, asa GL2(Zp ⊗Z OF )-subrepresentation, a principal series type σ, which is a repre-sentation of GL2(OF /p) induced from a character of the Borel subgroup. In thepaper [16], Breuil gave a conjectural description of the isomorphism class of thelattice σ◦ := σ ∩ H0

m, purely in terms of the restriction to p of ρ◦. Under mildassumptions on ρm, this was proved in [35]. This gives some evidence towards thepossibility that (V ∨ ⊗L πp)◦ may be of a purely local nature.

3.3.4. Compact unitary groups. In [27], we apply Taylor–Wiles patching topass from completed cohomology over a unitary group that is compact at infinityto a G-representation on a module M∞ over the ring R∞ obtained by adjoininga certain number of formal variables to a local deformation ring. More precisely:for any finite extension K of Qp, any n such that p - 2n, and any representationr : GK → GLn(Fp) that admits a potentially crystalline lift which is potentiallydiagonalizable (in the sense of [4]), we can (by [33, Cor. A.7]) choose a unitary groupG and Hecke maximal ideal m so that G := G(Qp) is isomorphic to a product ofcopies of GLn(K), and such that the restriction of ρm to GK is equal to r. The G-representation M∞, which is a module over Rr[[x1, . . . , xn]] for some some n ≥ 0,is then obtained by patching the completed cohomology for G.

The module M∞ satisfies several of the desiderata of (3.2.2): it is finitely gen-erated over R∞[[Kp]], and the modules Homcont

Kp(M,W )d (for Kp-invariant lattices

W in algebraic representations) are supported on a union of components of theappropriate potentially semistable loci. Many questions about M∞ remain open,however: whether the support of M∞ equals the entirety of Spec R∞; whether allpotentially semistable points are contained in the support of Homcont

Kp(M,W )d (for

an appropriate choice of W and Kp); and whether M∞ is in fact of a purely localnature.

One further property of M∞ is that it is projective in the category of profi-nite topological O[[Kp]]-modules. In [37], we hope to show that M∞ is in fact offull support on Spec Rr[[x1, . . . , xn]] in many cases, and thus extend the methoddescribed in (3.1.3) to deduce density results for potentially semistable represen-tations in local deformation spaces.

24 Matthew Emerton

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Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago,IL 60637

E-mail: [email protected]

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